Try to determine the total possible telephone numbers in the U.S. using
the following: Area codes are three digits in which the middle digit is
either 0 or 1 and the first digit can be anything but 0 or 1. No three
exchange digits can be the same as the preceding area code.

An ambiguous combinatorics question confounds a student and his classmates
— along with the teacher who made it up. Doctor Peterson troubleshoots the
problem's possible cases, and further sets up and solves a simpler problem,
emphasizing throughout the importance of precise wording.

A variation on the classic Lights-Out Puzzle in which pushing any
button in a 6x6 grid changes the state of that button and all others
in the same row or column. The goal is to maximize the number of
buttons in a given state.

Choose 5 numbers from 1-40 and match 3 randomly picked numbers from
1-40 to win. How many different combinations of 5 numbers are there in
40 and how many different guesses of 5 would it take to cover every
possible outcome of three numbers?

I need to know the odds on the UK National Lottery: punters choose 6
different numbers between 1 and 49. How many permutations? How can I
calculate the reduced number of permutations to reflect the fact that the
order of the chosen numbers is not relevant?